AI Chatbot Design

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  Reverse correlation technique

  The reverse correlation technique is a data driven study method used primarily in psychological and neurophysiological research. This method earned its name from its origins in neurophysiology, where cross-correlations between white noise stimuli and sparsely occurring neuronal spikes could be computed quicker when only computing it for segments preceding the spikes. The term has since been adopted in psychological experiments that usually do not analyze the temporal dimension, but also present noise to human participants. In contrast to the original meaning, the term is here thought to reflect that the standard psychological practice of presenting stimuli of defined categories to the participants is "reversed": Instead, the participant's mental representations of categories are estimated from interactions of the presented noise and the behavioral responses. It is used to create composite pictures of individual and/or group mental representations of various items (e.g. faces, bodies, and the self) that depict characteristics of said items (e.g. trustworthiness and self-body image). This technique is helpful when evaluating the mental representations of those with and without mental illnesses. == Terms == This technique utilizes spike-triggered average to explain what areas of signal and noise in an image are valuable for the given research question. Signal is information used to produce objects of value that help explain and connect the world around us. Noise is commonly referred to as unwanted signal that obscures the information that the signal is trying to present. Most importantly for reverse correlation studies, noise is randomly varying information. To determine the areas of importance using reverse correlation, noise is applied to a base image and then evaluated by observers. A base image is any image void of noise that relates to the research question. A base image that has noise superimposed on top is the stimuli that is presented to and evaluated by participants. Each time a new set of stimuli is presented to a participant, this is known as a trial. After a participant has responded to hundreds to thousands of trials, a researcher is ready to create a classification image. A classification image (abbreviated as "CI" in some studies) is a single image that represents the average noise patterns in the images selected by participants. A classification image can also be computed for groups by averaging the individuals’ classification images. These classification images are what researchers use to interpret the data and draw conclusions. As a whole, the reverse correlation method is a process that results in a composite image (from an individual or group) that can be used to estimate and interpret mental representations. == Basic study layout == The reverse correlation method is typically executed as an in-lab computer experiment. This method follows four broad steps. Each of the following steps are described in greater detail below. After creating a research question and determining that the reverse correlation method is the most suitable technique to answer the question, a researcher must (1) design randomly varying stimuli. After the stimuli have been prepared, a researcher should (2) collect data from participants who will see and respond to approximately 300 -1,000 trials. Each trial will either consist of one or two images (side by side) derived from the same base image with noise superimposed on top. Participant responses will depend on the chosen study design; if a researcher presents only one image at a time, participants rate the image on a 4pt scale, but when two images are shown, the participant is asked to choose which best aligns with the given category (e.g. choose the image that looks the most aggressive). Once all of the data is collected, the researcher will (3) compute classification images for each participant and using those images compute group classification images. Finally, with the classification images available, the researcher will (4) evaluate the images and draw conclusions about their results. === Step 1: making stimuli === When designing the stimuli for a reverse correlation study, the two primary factors that one should consider are (1) the base image and (2) the noise that will be used. While not all bases are images per se, the majority are and for this reason the base is typically referred to as a base image. The base image should represent whatever the research question is addressing. For example, if you are interested in peoples’ mental representations of Chinese people, it would not make sense to use a base image of a Spanish or Caucasian person. Again, if you are interested in the mental representations of male vocal patterns, it would make the most sense to use a base vocal pattern that has been produced by a male. Having a base is important because it provides a kind of anchor for participants to work from. When there is no base image, the number of trials that are required increases dramatically, thus making it harder to collect data. While there are studies that have excluded a base image, (e.g. the S study), for more elaborate and nuanced research questions, it is important to have a base image that is a fair representation of what participants are being asked to categorize. Photographs of faces are generally the most popular base image. Although the reverse correlation method is capable of investigating a wide variety of research questions, the most common application of the method is for evaluating faces on a single trait. Reverse correlation studies that address evaluations of the face are sometimes referred to as being a face space reverse correlation model (FSRCM). Thankfully, there are existing databases for face images of varying demographics and emotion that work well as base images. The reverse correlation method can also be used to help researchers identify what areas of an image (e.g. the areas on the face) have diagnostic value. In order to identify these areas of value, researchers start by minimizing the space a participant can pull information from. By imposing a “mask” on an image (e.g. blur an image while leaving random areas un-blurred), this reduces the information individuals might see, and forces them to focus on certain areas. Then, if/when participants are able to correctly identify an image with a trait repeatedly, we can draw conclusions about what areas have diagnostic value. While faces and visual stimuli are the most popular, this is not the only stimuli that can be used in a reverse correlation study. This method was originally designed for auditory stimuli which allows researchers to investigate how perceivers interpret auditory information and create trait based attributions to different sound patterns. For example, by segmenting a vocal recording of a single word (total sound time 426 ms) into six segments (71 ms each), and varying each segment's pitch using Gaussian distributions, researchers were able to uncover what vocal patterns people associated with certain traits. Specifically, this study investigated how listeners rated sound clips of the word “really” as sounding more interrogative (i.e. like the more common reverse correlation studies this study had participants listen to two sound clips per trial, choose which fit the category the best, and then created an average of the pitch contours). Beyond face and auditory perception, research utilizing the reverse correlation method has expanded to investigate how individuals see three-dimensional objects in images with noise (but no signal). After selecting your base image, regardless of what the image is, it is helpful to apply a Gaussian blur to smooth noise in the image. While noise will be applied later, it is helpful to reduce existing noise in the photo before applying your chosen noise. There are three primary choices when it comes to noise: white noise, sine-wave noise, and Gabor noise. The latter two of these constrain the configurations that the noise can have, and because of this white noise is usually the most commonly used. Regardless of the type of noise that is chosen, it is crucial that the noise randomly varies. === Step 2: data collection === Once the stimuli for the study has been developed, the researcher must make a few decisions before actually collecting the data. The researcher must come to a conclusion on how many stimuli will be presented at a time and how many trials the participants will see. In terms of stimuli presentation, a researcher can choose from either a 2-Image Forced Choice (2IFC) or a 4-Alternative Forced Choice (4AFC). The 2IFC presents two images at once (side by side) and requires participants to choose between the two on a specified category (e.g. which image looks the most like a male). Typically the noise from the left image is the mathematical inverse of the noise from the right image. This method was developed to better answer questions that could n

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  Information gain ratio

  In decision tree learning, information gain ratio is a ratio of information gain to the intrinsic information. It was proposed by Ross Quinlan, to reduce a bias towards multi-valued attributes by taking the number and size of branches into account when choosing an attribute. Information gain is also known as mutual information. == Information gain calculation == Information gain is the reduction in entropy produced from partitioning a set with attributes a {\displaystyle a} and finding the optimal candidate that produces the highest value: IG ( T , a ) = H ( T ) − H ( T | a ) , {\displaystyle {\text{IG}}(T,a)=\mathrm {H} {(T)}-\mathrm {H} {(T|a)},} where T {\displaystyle T} is a random variable and H ( T | a ) {\displaystyle \mathrm {H} {(T|a)}} is the entropy of T {\displaystyle T} given the value of attribute a {\displaystyle a} . The information gain is equal to the total entropy for an attribute if for each of the attribute values a unique classification can be made for the result attribute. In this case the relative entropies subtracted from the total entropy are 0. == Split information calculation == The split information value for a test is defined as follows: SplitInformation ( X ) = − ∑ i = 1 n N ( x i ) N ( x ) ∗ log ⁡ 2 N ( x i ) N ( x ) {\displaystyle {\text{SplitInformation}}(X)=-\sum _{i=1}^{n}{{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}\log {_{2}}{\frac {\mathrm {N} (x_{i})}{\mathrm {N} (x)}}}} where X {\displaystyle X} is a discrete random variable with possible values x 1 , x 2 , . . . , x i {\displaystyle {x_{1},x_{2},...,x_{i}}} and N ( x i ) {\displaystyle N(x_{i})} being the number of times that x i {\displaystyle x_{i}} occurs divided by the total count of events N ( x ) {\displaystyle N(x)} where x {\displaystyle x} is the set of events. The split information value is a positive number that describes the potential worth of splitting a branch from a node. This in turn is the intrinsic value that the random variable possesses and will be used to remove the bias in the information gain ratio calculation. == Information gain ratio calculation == The information gain ratio is the ratio between the information gain and the split information value: IGR ( T , a ) = IG ( T , a ) / SplitInformation ( T ) {\displaystyle {\text{IGR}}(T,a)={\text{IG}}(T,a)/{\text{SplitInformation}}(T)} IGR ( T , a ) = − ∑ i = 1 n P ( T ) log ⁡ P ( T ) − ( − ∑ i = 1 n P ( T | a ) log ⁡ P ( T | a ) ) − ∑ i = 1 n N ( t i ) N ( t ) ∗ log ⁡ 2 N ( t i ) N ( t ) {\displaystyle {\text{IGR}}(T,a)={\frac {-\sum _{i=1}^{n}{\mathrm {P} (T)\log \mathrm {P} (T)}-(-\sum _{i=1}^{n}{\mathrm {P} (T|a)\log \mathrm {P} (T|a)})}{-\sum _{i=1}^{n}{{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}\log {_{2}}{\frac {\mathrm {N} (t_{i})}{\mathrm {N} (t)}}}}}} == Example == Using weather data published by Fordham University, the table was created below: Using the table above, one can find the entropy, information gain, split information, and information gain ratio for each variable (outlook, temperature, humidity, and wind). These calculations are shown in the tables below: Using the above tables, one can deduce that Outlook has the highest information gain ratio. Next, one must find the statistics for the sub-groups of the Outlook variable (sunny, overcast, and rainy), for this example one will only build the sunny branch (as shown in the table below): One can find the following statistics for the other variables (temperature, humidity, and wind) to see which have the greatest effect on the sunny element of the outlook variable: Humidity was found to have the highest information gain ratio. One will repeat the same steps as before and find the statistics for the events of the Humidity variable (high and normal): Since the play values are either all "No" or "Yes", the information gain ratio value will be equal to 1. Also, now that one has reached the end of the variable chain with Wind being the last variable left, they can build an entire root to leaf node branch line of a decision tree. Once finished with reaching this leaf node, one would follow the same procedure for the rest of the elements that have yet to be split in the decision tree. This set of data was relatively small, however, if a larger set was used, the advantages of using the information gain ratio as the splitting factor of a decision tree can be seen more. == Advantages == Information gain ratio biases the decision tree against considering attributes with a large number of distinct values. For example, suppose that we are building a decision tree for some data describing a business's customers. Information gain ratio is used to decide which of the attributes are the most relevant. These will be tested near the root of the tree. One of the input attributes might be the customer's telephone number. This attribute has a high information gain, because it uniquely identifies each customer. Due to its high amount of distinct values, this will not be chosen to be tested near the root. == Disadvantages == Although information gain ratio solves the key problem of information gain, it creates another problem. If one is considering an amount of attributes that have a high number of distinct values, these will never be above one that has a lower number of distinct values. == Difference from information gain == Information gain's shortcoming is created by not providing a numerical difference between attributes with high distinct values from those that have less. Example: Suppose that we are building a decision tree for some data describing a business's customers. Information gain is often used to decide which of the attributes are the most relevant, so they can be tested near the root of the tree. One of the input attributes might be the customer's credit card number. This attribute has a high information gain, because it uniquely identifies each customer, but we do not want to include it in the decision tree: deciding how to treat a customer based on their credit card number is unlikely to generalize to customers we haven't seen before. Information gain ratio's strength is that it has a bias towards the attributes with the lower number of distinct values. Below is a table describing the differences of information gain and information gain ratio when put in certain scenarios.

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  Nearest neighbor search

  Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M {\displaystyle q\in M} , find the closest point in S to q. Donald Knuth in volume 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. == Applications == The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Vector databases, where nearest-neighbor lookup over embeddings is used to retrieve semantically similar records Retrieval-augmented generation systems, where nearest-neighbor retrieval over embeddings is used to fetch candidate passages or documents before generation Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning == Methods == Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. === Exact methods === ==== Linear search ==== The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. ==== Space partitioning ==== Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. === Approximation methods === An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most c {\displaystyle c} times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. ==== Greedy search in proximity neighborhood graphs ==== Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs G ( V , E ) {\displaystyle G(V,E)} in which every point x i ∈ S {\displaystyle x_{i}\in S} is uniquely associated with vertex v i ∈ V {\displaystyle v_{i}\in V} . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph G ( V , E ) {\displaystyle G(V,E)} . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex v i ∈ V {\displaystyle v_{i}\in V} by computing the distances from the query q to each vertex of its neighborhood { v j : ( v i , v j ) ∈ E } {\displaystyle \{v_{j}:(v_{i},v_{j})\in E\}} , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet syst

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  Optimal discriminant analysis and classification tree analysis

  Optimal Discriminant Analysis (ODA) and the related classification tree analysis (CTA) are exact statistical methods that maximize predictive accuracy. For any specific sample and exploratory or confirmatory hypothesis, optimal discriminant analysis (ODA) identifies the statistical model that yields maximum predictive accuracy, assesses the exact Type I error rate, and evaluates potential cross-generalizability. Optimal discriminant analysis may be applied to > 0 dimensions, with the one-dimensional case being referred to as UniODA and the multidimensional case being referred to as MultiODA. Optimal discriminant analysis is an alternative to ANOVA (analysis of variance) and regression analysis.

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  Conference app

  A conference app, also known as an event app or meeting app, is a mobile app developed to help attendees and meeting planners manage their conference experience. It typically includes conference proceedings and venue information, allowing users to create personalized schedules and engage with other users. A conference app can be a native app or web-based. In recent years, conference apps have gained in popularity as a sustainable solution for event management by reducing paper produced by printed materials. Advanced features often include real-time notifications for updates or changes, integration with virtual meeting platforms for hybrid or fully online events, and analytics tools for organizers to measure attendance and engagement. Additionally, some apps support sponsorship and exhibitor features, enabling businesses to showcase their products or services directly within the app.

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  Sum of absolute transformed differences

  The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.

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  Quickprop

  Quickprop is an iterative method for determining the minimum of the loss function of an artificial neural network, following an algorithm inspired by the Newton's method. Sometimes, the algorithm is classified to the group of the second order learning methods. It follows a quadratic approximation of the previous gradient step and the current gradient, which is expected to be close to the minimum of the loss function, under the assumption that the loss function is locally approximately square, trying to describe it by means of an upwardly open parabola. The minimum is sought in the vertex of the parabola. The procedure requires only local information of the artificial neuron to which it is applied. The k {\displaystyle k} -th approximation step is given by: Δ ( k ) w i j = Δ ( k − 1 ) w i j ( ∇ i j E ( k ) ∇ i j E ( k − 1 ) − ∇ i j E ( k ) ) {\displaystyle \Delta ^{(k)}\,w_{ij}=\Delta ^{(k-1)}\,w_{ij}\left({\frac {\nabla _{ij}\,E^{(k)}}{\nabla _{ij}\,E^{(k-1)}-\nabla _{ij}\,E^{(k)}}}\right)} Where w i j {\displaystyle w_{ij}} is the weight of input i {\displaystyle i} of neuron j {\displaystyle j} , and E {\displaystyle E} is the loss function. The Quickprop algorithm is an implementation of the error backpropagation algorithm, but the network can behave chaotically during the learning phase due to large step sizes.

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  ID3 algorithm

  In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.

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  Jive (software)

  Jive (formerly known as Clearspace, then Jive SBS, then Jive Engage) is a commercial Java EE-based Enterprise 2.0 collaboration and knowledge management tool produced by Jive Software. It was first released as "Clearspace" in 2006, then renamed SBS (for "Social Business Software") in March 2009, then renamed "Jive Engage" in 2011, and renamed simply to "Jive" in 2012. Jive integrates the functionality of online communities, microblogging, social networking, discussion forums, blogs, wikis, and IM under one unified user interface. Content placed into any of the systems (blog, wiki, documentation, etc.) can be found through a common search interface. Other features include RSS capability, email integration, a reputation and reward system for participation, personal user profiles, JAX-WS web service interoperability, and integration with the Spring Framework. The product is a pure-Java server-side web application and will run on any platform where Java (JDK 1.5 or higher) is installed. It does not require a dedicated server - users have reported successful deployment in both shared environments and multiple machine clusters. As of Jive 8, released March 30, 2015, there is a Jive-n version which is for internal use (hosted by the consumer or hosted by Jive as a service) and a Jive-x version which is an external version hosted as a service. Jive no longer supports wiki markup language. == Server requirements for Jive 8-n == The following are the server requirements for Jive 8-n Operating systems: RHEL version 6 or 7 for x86_64, CentOS version 6 or 7 for x86_64 or SuSE Enterprise Linux Server (SLES) 11 and 12 for x86_64 Application Servers: Jive ships with its own embedded Apache HTTPD and Tomcat servers as part of the install package. It is not possible to deploy the application onto other appservers. Databases: MySQL (5.1, 5.5, 5.6) Oracle (11gR2, 12c) Postgres (9.0, 9.1, 9.2, 9.3, 9.4 - 9.2 or higher recommended) Microsoft SQL Server (2008R2, 2012, 2014) Environment: Jive recommends a server with at least 4GB of RAM and a dual-core 2 GHz processor with x86_64 architecture The product integrates with an LDAP repository or Active Directory For optimal deployment with a large community Jive Software recommends: using dedicated cache and document-conversion servers hosting the application and database servers separately == Releases == Jive 8, released on March 30, 2015 Jive 7, released in October 2013 Jive 9.0.x, released in November 2016 Jive 9, released in November 2016, supported now

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  Information Harvesting

  Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.

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  Generalized multidimensional scaling

  Generalized multidimensional scaling (GMDS) is an extension of metric multidimensional scaling, in which the target space is non-Euclidean. When the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another. GMDS is an emerging research direction. Currently, main applications are recognition of deformable objects (e.g. for three-dimensional face recognition) and texture mapping.

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  Dominance-based rough set approach

  The dominance-based rough set approach (DRSA) is an extension of rough set theory for multi-criteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński. The main change compared to the classical rough sets is the substitution for the indiscernibility relation by a dominance relation, which permits one to deal with inconsistencies typical to consideration of criteria and preference-ordered decision classes. == Multicriteria classification (sorting) == Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class. Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in real-life application when ordinal and monotone properties follow from the domain knowledge about the problem. As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). Each student is described by three criteria: level in Physics, Mathematics and Literature, each taking one of three possible values bad, medium and good. Criteria are preference-ordered and improving the level from one of the subjects should not result in worse global evaluation (class). As a more serious example, consider classification of bank clients, from the viewpoint of bankruptcy risk, into classes safe and risky. This may involve such characteristics as "return on equity (ROE)", "return on investment (ROI)" and "return on sales (ROS)". The domains of these attributes are not simply ordered but involve a preference order since, from the viewpoint of bank managers, greater values of ROE, ROI or ROS are better for clients being analysed for bankruptcy risk . Thus, these attributes are criteria. Neglecting this information in knowledge discovery may lead to wrong conclusions. == Data representation == === Decision table === In DRSA, data are often presented using a particular form of decision table. Formally, a DRSA decision table is a 4-tuple S = ⟨ U , Q , V , f ⟩ {\displaystyle S=\langle U,Q,V,f\rangle } , where U {\displaystyle U\,\!} is a finite set of objects, Q {\displaystyle Q\,\!} is a finite set of criteria, V = ⋃ q ∈ Q V q {\displaystyle V=\bigcup {}_{q\in Q}V_{q}} where V q {\displaystyle V_{q}\,\!} is the domain of the criterion q {\displaystyle q\,\!} and f : U × Q → V {\displaystyle f\colon U\times Q\to V} is an information function such that f ( x , q ) ∈ V q {\displaystyle f(x,q)\in V_{q}} for every ( x , q ) ∈ U × Q {\displaystyle (x,q)\in U\times Q} . The set Q {\displaystyle Q\,\!} is divided into condition criteria (set C ≠ ∅ {\displaystyle C\neq \emptyset } ) and the decision criterion (class) d {\displaystyle d\,\!} . Notice, that f ( x , q ) {\displaystyle f(x,q)\,\!} is an evaluation of object x {\displaystyle x\,\!} on criterion q ∈ C {\displaystyle q\in C} , while f ( x , d ) {\displaystyle f(x,d)\,\!} is the class assignment (decision value) of the object. An example of decision table is shown in Table 1 below. === Outranking relation === It is assumed that the domain of a criterion q ∈ Q {\displaystyle q\in Q} is completely preordered by an outranking relation ⪰ q {\displaystyle \succeq _{q}} ; x ⪰ q y {\displaystyle x\succeq _{q}y} means that x {\displaystyle x\,\!} is at least as good as (outranks) y {\displaystyle y\,\!} with respect to the criterion q {\displaystyle q\,\!} . Without loss of generality, we assume that the domain of q {\displaystyle q\,\!} is a subset of reals, V q ⊆ R {\displaystyle V_{q}\subseteq \mathbb {R} } , and that the outranking relation is a simple order between real numbers ≥ {\displaystyle \geq \,\!} such that the following relation holds: x ⪰ q y ⟺ f ( x , q ) ≥ f ( y , q ) {\displaystyle x\succeq _{q}y\iff f(x,q)\geq f(y,q)} . This relation is straightforward for gain-type ("the more, the better") criterion, e.g. company profit. For cost-type ("the less, the better") criterion, e.g. product price, this relation can be satisfied by negating the values from V q {\displaystyle V_{q}\,\!} . === Decision classes and class unions === Let T = { 1 , … , n } {\displaystyle T=\{1,\ldots ,n\}\,\!} . The domain of decision criterion, V d {\displaystyle V_{d}\,\!} consist of n {\displaystyle n\,\!} elements (without loss of generality we assume V d = T {\displaystyle V_{d}=T\,\!} ) and induces a partition of U {\displaystyle U\,\!} into n {\displaystyle n\,\!} classes Cl = { C l t , t ∈ T } {\displaystyle {\textbf {Cl}}=\{Cl_{t},t\in T\}} , where C l t = { x ∈ U : f ( x , d ) = t } {\displaystyle Cl_{t}=\{x\in U\colon f(x,d)=t\}} . Each object x ∈ U {\displaystyle x\in U} is assigned to one and only one class C l t , t ∈ T {\displaystyle Cl_{t},t\in T} . The classes are preference-ordered according to an increasing order of class indices, i.e. for all r , s ∈ T {\displaystyle r,s\in T} such that r ≥ s {\displaystyle r\geq s\,\!} , the objects from C l r {\displaystyle Cl_{r}\,\!} are strictly preferred to the objects from C l s {\displaystyle Cl_{s}\,\!} . For this reason, we can consider the upward and downward unions of classes, defined respectively, as: C l t ≥ = ⋃ s ≥ t C l s C l t ≤ = ⋃ s ≤ t C l s t ∈ T {\displaystyle Cl_{t}^{\geq }=\bigcup _{s\geq t}Cl_{s}\qquad Cl_{t}^{\leq }=\bigcup _{s\leq t}Cl_{s}\qquad t\in T} == Main concepts == === Dominance === We say that x {\displaystyle x\,\!} dominates y {\displaystyle y\,\!} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted by x D p y {\displaystyle xD_{p}y\,\!} , if x {\displaystyle x\,\!} is better than y {\displaystyle y\,\!} on every criterion from P {\displaystyle P\,\!} , x ⪰ q y , ∀ q ∈ P {\displaystyle x\succeq _{q}y,\,\forall q\in P} . For each P ⊆ C {\displaystyle P\subseteq C} , the dominance relation D P {\displaystyle D_{P}\,\!} is reflexive and transitive, i.e. it is a partial pre-order. Given P ⊆ C {\displaystyle P\subseteq C} and x ∈ U {\displaystyle x\in U} , let D P + ( x ) = { y ∈ U : y D p x } {\displaystyle D_{P}^{+}(x)=\{y\in U\colon yD_{p}x\}} D P − ( x ) = { y ∈ U : x D p y } {\displaystyle D_{P}^{-}(x)=\{y\in U\colon xD_{p}y\}} represent P-dominating set and P-dominated set with respect to x ∈ U {\displaystyle x\in U} , respectively. === Rough approximations === The key idea of the rough set philosophy is approximation of one knowledge by another knowledge. In DRSA, the knowledge being approximated is a collection of upward and downward unions of decision classes and the "granules of knowledge" used for approximation are P-dominating and P-dominated sets. The P-lower and the P-upper approximation of C l t ≥ , t ∈ T {\displaystyle Cl_{t}^{\geq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} and P ¯ ( C l t ≥ ) {\displaystyle {\overline {P}}(Cl_{t}^{\geq })} , respectively, are defined as: P _ ( C l t ≥ ) = { x ∈ U : D P + ( x ) ⊆ C l t ≥ } {\displaystyle {\underline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{+}(x)\subseteq Cl_{t}^{\geq }\}} P ¯ ( C l t ≥ ) = { x ∈ U : D P − ( x ) ∩ C l t ≥ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{-}(x)\cap Cl_{t}^{\geq }\neq \emptyset \}} Analogously, the P-lower and the P-upper approximation of C l t ≤ , t ∈ T {\displaystyle Cl_{t}^{\leq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≤ ) {\displaystyle {\underline {P}}(Cl_{t}^{\leq })} and P ¯ ( C l t ≤ ) {\displaystyle {\overline {P}}(Cl_{t}^{\leq })} , respectively, are defined as: P _ ( C l t ≤ ) = { x ∈ U : D P − ( x ) ⊆ C l t ≤ } {\displaystyle {\underline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{-}(x)\subseteq Cl_{t}^{\leq }\}} P ¯ ( C l t ≤ ) = { x ∈ U : D P + ( x ) ∩ C l t ≤ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{+}(x)\cap Cl_{t}^{\leq }\neq \emptyset \}} Lower approximations group the objects which certainly belong to class union C l t ≥ {\displaystyle Cl_{t}^{\geq }} (respectively C l t ≤ {\displaystyle Cl_{t}^{\leq }} ). This certainty comes from the fact, that object x ∈ U {\displaystyle x\in U} belongs to the lower approximation P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} (respectively P _ ( C l t ≤ ) {\displaystyle {\underl

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  Automated essay scoring

  Automated essay scoring (AES) is the use of specialized computer programs to assign grades to essays written in an educational setting. It is a form of educational assessment and an application of natural language processing. Its objective is to classify a large set of textual entities into a small number of discrete categories, corresponding to the possible grades, for example, the numbers 1 to 6. Therefore, it can be considered a problem of statistical classification. Several factors have contributed to a growing interest in AES. Among them are cost, accountability, standards, and technology. Rising education costs have led to pressure to hold the educational system accountable for results by imposing standards. The advance of information technology promises to measure educational achievement at reduced cost. The use of AES for high-stakes testing in education has generated significant backlash, with opponents pointing to research that computers cannot yet grade writing accurately and arguing that their use for such purposes promotes teaching writing in reductive ways (i.e. teaching to the test). == History == Most historical summaries of AES trace the origins of the field to the work of Ellis Batten Page. In 1966, he argued for the possibility of scoring essays by computer, and in 1968 he published his successful work with a program called Project Essay Grade (PEG). Using the technology of that time, computerized essay scoring would not have been cost-effective, so Page abated his efforts for about two decades. Eventually, Page sold PEG to Measurement Incorporated. By 1990, desktop computers had become so powerful and so widespread that AES was a practical possibility. As early as 1982, a UNIX program called Writer's Workbench was able to offer punctuation, spelling and grammar advice. In collaboration with several companies (notably Educational Testing Service), Page updated PEG and ran some successful trials in the early 1990s. Peter Foltz and Thomas Landauer developed a system using a scoring engine called the Intelligent Essay Assessor (IEA). IEA was first used to score essays in 1997 for their undergraduate courses. It is now a product from Pearson Educational Technologies and used for scoring within a number of commercial products and state and national exams. IntelliMetric is Vantage Learning's AES engine. Its development began in 1996. It was first used commercially to score essays in 1998. Educational Testing Service offers "e-rater", an automated essay scoring program. It was first used commercially in February 1999. Jill Burstein was the team leader in its development. ETS's Criterion Online Writing Evaluation Service uses the e-rater engine to provide both scores and targeted feedback. Lawrence Rudner has done some work with Bayesian scoring, and developed a system called BETSY (Bayesian Essay Test Scoring sYstem). Some of his results have been published in print or online, but no commercial system incorporates BETSY as yet. Under the leadership of Howard Mitzel and Sue Lottridge, Pacific Metrics developed a constructed response automated scoring engine, CRASE. Currently utilized by several state departments of education and in a U.S. Department of Education-funded Enhanced Assessment Grant, Pacific Metrics’ technology has been used in large-scale formative and summative assessment environments since 2007. Measurement Inc. acquired the rights to PEG in 2002 and has continued to develop it. In 2012, the Hewlett Foundation sponsored a competition on Kaggle called the Automated Student Assessment Prize (ASAP). 201 challenge participants attempted to predict, using AES, the scores that human raters would give to thousands of essays written to eight different prompts. The intent was to demonstrate that AES can be as reliable as human raters, or more so. The competition also hosted a separate demonstration among nine AES vendors on a subset of the ASAP data. Although the investigators reported that the automated essay scoring was as reliable as human scoring, this claim was not substantiated by any statistical tests because some of the vendors required that no such tests be performed as a precondition for their participation. Moreover, the claim that the Hewlett Study demonstrated that AES can be as reliable as human raters has since been strongly contested, including by Randy E. Bennett, the Norman O. Frederiksen Chair in Assessment Innovation at the Educational Testing Service. Some of the major criticisms of the study have been that five of the eight datasets consisted of paragraphs rather than essays, four of the eight data sets were graded by human readers for content only rather than for writing ability, and that rather than measuring human readers and the AES machines against the "true score", the average of the two readers' scores, the study employed an artificial construct, the "resolved score", which in four datasets consisted of the higher of the two human scores if there was a disagreement. This last practice, in particular, gave the machines an unfair advantage by allowing them to round up for these datasets. In 1966, Page hypothesized that, in the future, the computer-based judge will be better correlated with each human judge than the other human judges are. Despite criticizing the applicability of this approach to essay marking in general, this hypothesis was supported for marking free text answers to short questions, such as those typical of the British GCSE system. Results of supervised learning demonstrate that the automatic systems perform well when marking by different human teachers is in good agreement. Unsupervised clustering of answers showed that excellent papers and weak papers formed well-defined clusters, and the automated marking rule for these clusters worked well, whereas marks given by human teachers for the third cluster ('mixed') can be controversial, and the reliability of any assessment of works from the 'mixed' cluster can often be questioned (both human and computer-based). == Different dimensions of essay quality == According to a recent survey, modern AES systems try to score different dimensions of an essay's quality in order to provide feedback to users. These dimensions include the following items: Grammaticality: following grammar rules Usage: using of prepositions, word usage Mechanics: following rules for spelling, punctuation, capitalization Style: word choice, sentence structure variety Relevance: how relevant of the content to the prompt Organization: how well the essay is structured Development: development of ideas with examples Cohesion: appropriate use of transition phrases Coherence: appropriate transitions between ideas Thesis Clarity: clarity of the thesis Persuasiveness: convincingness of the major argument == Procedure == From the beginning, the basic procedure for AES has been to start with a training set of essays that have been carefully hand-scored. The program evaluates surface features of the text of each essay, such as the total number of words, the number of subordinate clauses, or the ratio of uppercase to lowercase letters—quantities that can be measured without any human insight. It then constructs a mathematical model that relates these quantities to the scores that the essays received. The same model is then applied to calculate scores of new essays. Recently, one such mathematical model was created by Isaac Persing and Vincent Ng. which not only evaluates essays on the above features, but also on their argument strength. It evaluates various features of the essay, such as the agreement level of the author and reasons for the same, adherence to the prompt's topic, locations of argument components (major claim, claim, premise), errors in the arguments, cohesion in the arguments among various other features. In contrast to the other models mentioned above, this model is closer in duplicating human insight while grading essays. Due to the growing popularity of deep neural networks, deep learning approaches have been adopted for automated essay scoring, generally obtaining superior results, often surpassing inter-human agreement levels. The various AES programs differ in what specific surface features they measure, how many essays are required in the training set, and most significantly in the mathematical modeling technique. Early attempts used linear regression. Modern systems may use linear regression or other machine learning techniques often in combination with other statistical techniques such as latent semantic analysis and Bayesian inference. The automated essay scoring task has also been studied in the cross-domain setting using machine learning models, where the models are trained on essays written for one prompt (topic) and tested on essays written for another prompt. Successful approaches in the cross-domain scenario are based on deep neural networks or models that combine deep and shallow features. == Criteria for success == Any method of a

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  Generalized canonical correlation

  In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA. == Applications == The Helmert-Wolf blocking (HWB) method of estimating linear regression parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.

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  Swish function

  The swish function is a family of mathematical function defined as follows: swish β ⁡ ( x ) = x sigmoid ⁡ ( β x ) = x 1 + e − β x . {\displaystyle \operatorname {swish} _{\beta }(x)=x\operatorname {sigmoid} (\beta x)={\frac {x}{1+e^{-\beta x}}}.} where β {\displaystyle \beta } can be constant (usually set to 1) or trainable and "sigmoid" refers to the logistic function. The swish family was designed to smoothly interpolate between a linear function and the Rectified linear unit (ReLU) function. When considering positive values, Swish is a particular case of doubly parameterized sigmoid shrinkage function defined in . Variants of the swish function include Mish. == Special values == For β = 0, the function is linear: f(x) = x/2. For β = 1, the function is the Sigmoid Linear Unit (SiLU). For β = 1.702, the function approximates GeLU. With β → ∞, the function converges to ReLU. Thus, the swish family smoothly interpolates between a linear function and the ReLU function. Since swish β ⁡ ( x ) = swish 1 ⁡ ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , all instances of swish have the same shape as the default swish 1 {\displaystyle \operatorname {swish} _{1}} , zoomed by β {\displaystyle \beta } . One usually sets β > 0 {\displaystyle \beta >0} . When β {\displaystyle \beta } is trainable, this constraint can be enforced by β = e b {\displaystyle \beta =e^{b}} , where b {\displaystyle b} is trainable. swish 1 ⁡ ( x ) = x 2 + x 2 4 − x 4 48 + x 6 480 + O ( x 8 ) {\displaystyle \operatorname {swish} _{1}(x)={\frac {x}{2}}+{\frac {x^{2}}{4}}-{\frac {x^{4}}{48}}+{\frac {x^{6}}{480}}+O\left(x^{8}\right)} swish 1 ⁡ ( x ) = x 2 tanh ⁡ ( x 2 ) + x 2 swish 1 ⁡ ( x ) + swish − 1 ⁡ ( x ) = x tanh ⁡ ( x 2 ) swish 1 ⁡ ( x ) − swish − 1 ⁡ ( x ) = x {\displaystyle {\begin{aligned}\operatorname {swish} _{1}(x)&={\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)+{\frac {x}{2}}\\\operatorname {swish} _{1}(x)+\operatorname {swish} _{-1}(x)&=x\tanh \left({\frac {x}{2}}\right)\\\operatorname {swish} _{1}(x)-\operatorname {swish} _{-1}(x)&=x\end{aligned}}} == Derivatives == Because swish β ⁡ ( x ) = swish 1 ⁡ ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , it suffices to calculate its derivatives for the default case. swish 1 ′ ⁡ ( x ) = x + sinh ⁡ ( x ) 4 cosh 2 ⁡ ( x 2 ) + 1 2 {\displaystyle \operatorname {swish} _{1}'(x)={\frac {x+\sinh(x)}{4\cosh ^{2}\left({\frac {x}{2}}\right)}}+{\frac {1}{2}}} so swish 1 ′ ⁡ ( x ) − 1 2 {\displaystyle \operatorname {swish} _{1}'(x)-{\frac {1}{2}}} is odd. swish 1 ″ ⁡ ( x ) = 1 − x 2 tanh ⁡ ( x 2 ) 2 cosh 2 ⁡ ( x 2 ) {\displaystyle \operatorname {swish} _{1}''(x)={\frac {1-{\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)}{2\cosh ^{2}\left({\frac {x}{2}}\right)}}} so swish 1 ″ ⁡ ( x ) {\displaystyle \operatorname {swish} _{1}''(x)} is even. == History == SiLU was first proposed alongside the GELU in 2016, then again proposed in 2017 as the Sigmoid-weighted Linear Unit (SiL) in reinforcement learning. The SiLU/SiL was then again proposed as the SWISH over a year after its initial discovery, originally proposed without the learnable parameter β, so that β implicitly equaled 1. The swish paper was then updated to propose the activation with the learnable parameter β. In 2017, after performing analysis on ImageNet data, researchers from Google indicated that using this function as an activation function in artificial neural networks improves the performance, compared to ReLU and sigmoid functions. It is believed that one reason for the improvement is that the swish function helps alleviate the vanishing gradient problem during backpropagation.

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